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Gauss Test for Convergence

  1. Oct 5, 2011 #1
    1. The problem statement, all variables and given/known data

    I just got done proving Gauss' test, which is given in the book as:

    If there is an [itex]N\ge 1[/itex], an [itex]s>1[/itex], and an [itex]M>0[/itex] such that
    [tex]\frac{a_{n+1}}{a_n}=1 - \frac{A}{n} + \frac{f(n)}{n^s}[/tex]
    where [itex]|f(n)|\le M[/itex] for all n, then [itex]\sum a_n[/itex] converges if [itex]A>1[/itex] and diverges if [itex]A \le 1[/itex].

    This is equivalent to the many other forms I have found on the web. The next question asks to use this test to prove that the series

    [tex]\sum_{n=1} ^{\infty} \left( \frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot (2n)} \right)^k[/tex]

    converges if [itex]k>2[/itex] and diverges if [itex]k \le 2[/itex] using Gauss' test.



    2. Relevant equations



    3. The attempt at a solution


    OK, so much for the preamble. Here's my attempt:

    [tex]\frac{a_{n+1}}{a_n} = \left ( \frac {2n+1}{2n+2} \right ) ^ k[/tex]

    And that's it... I don't know how to put this into a form which corresponds in general to the form required for Gauss' test. I saw a similar problem online solved with the use of the fact that [itex]\left (\frac{n}{n+1} \right)^k \approx 1 - \frac{k}{n}[/itex] for large n, but the book has not covered anything like that (it introduced the ~ symbol, but not [itex]\approx[/itex]).
     
  2. jcsd
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